Fundamental Solutions for Pseudodifferential Operators, and Equations of Schrödinger Type

Abstract

This chapter aims to explore the connections between local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields. In the 50s Gel’fand and Shilov showed that fundamental solutions for certain types of partial differential operators with constant coefficients can be obtained by using local zeta functions [49]. The existence of fundamental solutions for general differential operators with constant coefficients was established by Atiyah [15] and Bernstein [18] using local zeta functions. A similar program can be carried out in the p-adic setting. In this chapter, we give a detailed proof of the existence of a fundamental solution for a pseudo-differential operator with a symbol of the form \(\left \vert \,f\right \vert _{p}^{\alpha }\), with f an arbitrary polynomial, see Theorem 134. This result was proved in [123], see also [120], by Zuniga-Galindo, here we present a complete proof, including a review of the method of analytic continuation of Gel’fand-Shilov, see [49].